A Machine Learning-Aided Equilibrium Model of VTSA Processes for Sorbents Screening Applied to CO2 Capture from Diluted Sources

The large design space of the sorbents’ structure and the associated capability of tailoring properties to match process requirements make adsorption-based technologies suitable candidates for improved CO2 capture processes. This is particularly of interest in novel, diluted, and ultradiluted separations as direct CO2 removal from the atmosphere. Here, we present an equilibrium model of vacuum temperature swing adsorption cycles that is suitable for large throughput sorbent screening, e.g., for direct air capture applications. The accuracy and prediction capabilities of the equilibrium model are improved by incorporating feed-forward neural networks, which are trained with data from rate-based models. This allows one, for example, to include the process productivity, a key performance indicator typically obtained in rate-based models. We show that the equilibrium model reproduces well the results of a sophisticated rate-based model in terms of both temperature and composition profiles for a fixed cycle as well as in terms of process optimization and sorbent comparison. Moreover, we apply the proposed equilibrium model to screen and identify promising sorbents from the large NIST/ARPA-E database; we do this for three different (ultra)diluted separation processes: direct air capture, yCO2 = 0.1%, and yCO2 = 1.0%. In all cases, the tool allows for a quick identification of the most promising sorbents and the computation of the associated performance indicators. Also, in this case, outcomes are very well in line with the 1D model results. The equilibrium model is available in the GitHub repository https://github.com/UU-ER/SorbentsScreening0D.

1 Modelling details

profile and time
The time for the blow-down step is mainly dependent on the vacuum pressure. By fitting simulation data from the 1D model, the following equation was received with the parameters listed in Table S1. Since the profile mainly varies with pressure and does not show a notable dependence on the density or temperature of the material, the fitting was carried out for a small dataset for one material (case s2 E-A) and different vacuum pressures over time. The pressure at each sub-step p k vac is calculated using the following equation The fitted parameters are listed in Table S2

Heating step: temperature profile and time
The training data of the neural network for determining the heating time can be found in Figure S1.
The temperature profile was received by fitting several profiles retrieved from simulations using the 1D model.
The profile is dependent on the desorption temperature and the heating time The fitted parameters are listed in Table S3.  Figure S1: Resulting training data (blue symbols) from the neural network together with the validation data (red) and the testing data (black).

Cooling step: profile
The time of the cooling step is fixed to 350 seconds, similar to the 1D model simulations. The temperature profile is calculated similar to the heating step, by fitting data from the 1D model. The fitting equation is with the parameters listed in Table S4.

Adsorption step
The total time of the adsorption step can be determined by including the air velocity u air and the geometry of the considered sorbent with M air being the molar mass of the air, A the column cross section and ρ air the density of the air.

Saturation level
Since we are considering an equilibrium model, the saturation at the end of the adsorption step or rather at the begin of the blow-down step is generally 100%. A more realistic picture of the saturation level gives the simulation with the detailed model, which shows a lower saturation level. Figure S2 shows the saturation level calculated for the Pareto points from the optimization of three different materials (the same optimization as used for the validation). The saturation level α was calculated for each grid point of the bed by calculating the actual loading referred to the full saturation at ambient conditions Figure S2: Saturation level from simulations using the 1D model. Figure S3: Training, testing and validation results.
with q n CO2 being the actual loading at the respective grid point and q * CO2 the loading at equilibrium. Using this data, a neural network was trained to determine the saturation level dependent on the particle density ρ p , the desorption temperature T des , the vacuum pressure p vac as well as the air volume flowV feed α = N N (ρ particle , T des , p vac ,V feed ) The training, testing and validation results can be found in Figure S3.

Isosteric heat of adsorption
The isosteric heat of adsorption (∆H ads ) represents the strength of the adsorbate-adsorbent interaction and is defined as the difference between the activation energy for adsorption and desorption. It can be calculated from where p CO2 is the partial pressure of CO 2 (Pa), T is the absolute temperature (K) and R is the universal gas constant. Blow-down step

Cooling step
Known: q 0 CO2 , q 0 H2O , y 0 CO2 , y 0 H2O , y 0 N2 , p, T Calculated: Nin, y k CO2 , y k H2O , y k N2 , Qcool 1.8 Details 1D model Table S8: Equations for 1D adsorption model [2] Total mass balance Mass transfer (Linear driving force model) Energy balance for the fixed bed Energy balance for the bed wall The decision variables for the detailed process simulation are listed in Table S9.

S13
The range of the resulting purity and recovery for both the 0D and the 1D model are shown in Figure S8.
The NIST/ARPA-E database, the database of novel and emerging adsorbents by the National Institute of Standards and Technology (NIST) includes data from published studies, both real and hypothetical. Since the database includes thousands of isotherm data files, not only for CO 2 adsorption, but also many other gases, it is important to first filter the materials. We use a script written in MATLAB to run through the materials filter the suitable adsorbents. The approach is graphically shown in Figure S9. During the filtering process, we are also excluding all data sets for which only one temperature is available. This is necessary for the following step, where the remaining isotherms are fitted by temperature dependent isotherm models. The fitting process itself includes several steps, which are shown in the flowchart in Figure S10. For the objective function during the fitting, the normalized standard deviation was applied, which is commonly used to fit isotherm models to experimental data [4][5][6][7]. It includes the adsorbed amount determined experimentally q exp , the amount adsorbed as predicted by the model q fit and the total number of experimental points N , and is calculated in the form of No Yes New initial parameters p0 Figure S10: Flowchart fitting.

S17
The error is expressed by with q exp being the mean of the experimental data. S18 Zeolite Na-LSX -21% +61% 10 Ca-X -27% +52% Table S14: Resulting ranking showing the 10 best performing materials for y CO2 = 0.1%. The values of the maximum productivity and the corresponding thermal energy consumption are given for the best performing adsorbent. For the remaining materials, the deviation to the best performing one is given in percentage. Zn-DABCO -26% +53% 10 MIL-101(Cr)-PEI-800 -71% -30% S19 Table S15: Resulting ranking showing the 10 best performing materials for y CO2 = 1.0%. The values of the maximum productivity and the corresponding thermal energy consumption are given for the best performing adsorbent. For the remaining materials, the deviation to the best performing one is given in percentage. Cr-MIL(101) -43% +318%

S22
The following three Tables S16, S17, and S18 list the resulting materials from the screening with different CO 2 concentrations for the case with the APDES-NFC isotherm for water.